Abstract: This paper introduces a statistical treatment of inverse problems constrained
by models with stochastic terms. The solution of the forward problem is given
by a distribution represented numerically by an ensemble of simulations. The
goal is to formulate the inverse problem, in particular the objective function,
to find the closest forward distribution (i.e., the output of the stochastic
forward problem) that best explains the distribution of the observations in a
certain metric. We use proper scoring rules, a concept employed in statistical
forecast verification, namely energy, variogram, and hybrid (i.e., combination
of the two) scores. We study the performance of the proposed formulation in the
context of two applications: a coefficient field inversion for subsurface flow
governed by an elliptic partial differential equation (PDE) with a stochastic
source and a parameter inversion for power grid governed by
differential-algebraic equations (DAEs). In both cases we show that the
variogram and the hybrid scores show better parameter inversion results than
does the energy score, whereas the energy score leads to better probabilistic
predictions.